Nonlinear Deformation Model of Crystal Media Allowing Martensite Transformations: Solution of Static Equations

Abstract

Mathematical methods are developed for solving the statics equations of a non-linear model of deforming a crystalline medium with a complex lattice that allows martensitic transformations. In the nonlinear theory, the deformation describes the vector of the acoustic mode U(t, x, y, z) and the vector of the optical mode u(t, x, y, z). They are found from a system of six related nonlinear equations. The vector of the acoustic mode U(t, x, y, z) is sought in the Papkovich–Neuberform. A system of six related nonlinear equations is transformed into a system of individual equations. The equations of the optical mode u(t, x, y, z) are reduced to one sine-Gordon equation with a variable coefficient (amplitude) in front of the sine and two Poisson equations. The definition of the acoustic mode is reduced to solving the scalar and vector Poisson equations. For a constant-amplitude optical mode, particular solutions were found. In the case of plane deformation, a class of doubly-periodic solutions is constructed, which are expressed in terms of Jacobi elliptic functions. The analysis of the solutions found is conducted. It is shown that the nonlinear theory describes the fragmentation of the crystalline medium, the formation of boundaries between fragments, phase transformations, the formation of defects and other deformation features that are realized in the field of high external force effects and which are not described by classical continuum mechanics.